Optimal. Leaf size=28 \[ -\frac{(c+d x)^2}{2 (a+b x)^2 (b c-a d)} \]
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Rubi [A] time = 0.0109377, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {24, 37} \[ -\frac{(c+d x)^2}{2 (a+b x)^2 (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 24
Rule 37
Rubi steps
\begin{align*} \int \frac{a c+(b c+a d) x+b d x^2}{(a+b x)^4} \, dx &=\frac{\int \frac{b^2 c+b^2 d x}{(a+b x)^3} \, dx}{b^2}\\ &=-\frac{(c+d x)^2}{2 (b c-a d) (a+b x)^2}\\ \end{align*}
Mathematica [A] time = 0.009448, size = 26, normalized size = 0.93 \[ -\frac{a d+b (c+2 d x)}{2 b^2 (a+b x)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 35, normalized size = 1.3 \begin{align*} -{\frac{-ad+bc}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}}-{\frac{d}{{b}^{2} \left ( bx+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06197, size = 51, normalized size = 1.82 \begin{align*} -\frac{2 \, b d x + b c + a d}{2 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6709, size = 81, normalized size = 2.89 \begin{align*} -\frac{2 \, b d x + b c + a d}{2 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.524705, size = 39, normalized size = 1.39 \begin{align*} - \frac{a d + b c + 2 b d x}{2 a^{2} b^{2} + 4 a b^{3} x + 2 b^{4} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17529, size = 32, normalized size = 1.14 \begin{align*} -\frac{2 \, b d x + b c + a d}{2 \,{\left (b x + a\right )}^{2} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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